An interesting class of feedback matrices, also explored by Jot
[217], is that of triangular
matrices. A basic fact from linear algebra
is that triangular matrices (either lower or upper triangular) have
all of their eigenvalues along the diagonal.4.13 For example, the
matrix

is lower triangular, and its eigenvalues are
for all values of
,
, and
.

It is important to note that not all triangular matrices are lossless.
For example, consider

It has two eigenvalues equal to 1, which looks lossless, but a quick
calculation shows that there is only one eigenvector,
. This
happens because this matrix is a Jordan block of order 2 corresponding
to the repeated eigenvalue
. A direct computation shows that